Math Analysis Permutations & Combinations

In this lesson, our instructor Vincent Selhorst-Jones teaches Permutations and Combinations. The introduction defines both terms and then you’ll learn about permutation formulas. Vincent goes over the permutation of n objects and r objects out of n. Distinguishable Permutations and combinations are also discussed before Vincent works through six practice problems on screen.

At the distinguishable permutations if we had the word ABBBC. There are six posibilities how we can arrange ABBBC. There are six pissibilities of how we can arrange BBBAC and so on. Is this why we divide it by 3! ?

Permutations & Combinations

A permutation is a way to order some set of objects. A combination is a way to choose some group of objects from a larger set of objects. In this case, we don't care about order, just which objects are selected.

Multiplying successive positive integers is shown by factorial notation. The symbol `!' is the symbol for factorial, and it works like this (where n ∈ ℕ):

n! = (n)(n−1)(n−2)…(3)(2)(1).

That is, it multiplies the number by every positive integer below the number. [We say `n!' as "n factorial".] To make certain math formulas easier to work with in the future, we define 0! = 1.

The number of permutations of n objects (in other words, how many different orders the objects can be put in) is equal to

n! = (n)(n−1)(n−2)…(3)(2)(1).

The number of ways to order r objects out of a set of n objects total is

n!

(n−r)!

.

This comes up often enough to have its own symbols: nPr or P(n,r).

If we're permuting a set where some of the objects are not distinguishable from one another (like the letters of a word), the number of distinguishable permutations is

n!

(n1!)(n2!)…(nk!)

,

where n is the total number of objects in the set and n1, …, nk, are the numbers of each object "type".

Given some set of n objects that we are going to choose r from (with no consideration for order), the number of possible combinations is

n!

r! ·(n−r)!

.

This comes up often enough to have its own symbols: nCr, (n || r), C(n,r).

Permutations & Combinations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.